The generalized traveling salesman problem solved with ant algorithms

Background Many combinatorial optimization problems are NP-hard, and the theory of NP-completeness has reduced hopes that NP-hard problems can be solved within polynomially bounded computation times (Dahlke 2008; Dunne 2008). Nevertheless, sub-optimal solutions are sometimes easy to find. Consequently, there is much interest in approximation and heuristic algorithms that can find near optimal solutions within reasonable running time. Heuristic algorithms are typically among the best strategies in terms of efficiency and solution quality for problems of realistic size and complexity. In contrast to individual heuristic algorithms that are designed to solve a specific problem, meta-heuristics are strategic problem solving frameworks that can be adapted to solve a wide variety of problems. Meta-heuristic algorithms are widely recognized as one of the most practical approaches for combinatorial optimization problems. The most representative meta-heuristics include genetic algorithms, simulated annealing, tabu search and ant colony. Useful references regarding meta-heuristic methods can be found in Glover and Kochenberger (2006). The generalized traveling salesman problem (GTSP) has been introduced in Laporte and Nobert (1983) and Noon and Bean (1991). The GTSP has several applications to location and telecommunication problems. More information on these problems and their applications can be found in Fischetti et al. (1997, 2007) and Laporte and Nobert (1983). Several approaches were considered for solving the GTSP: a branch-and-cut algorithm for Symmetric GTSP is described and analyzed in Fischetti et al. (1997), and Noon and Abstract A well known NP‐hard problem called the generalized traveling salesman problem (GTSP) is considered. In GTSP the nodes of a complete undirected graph are partitioned into clusters. The objective is to find a minimum cost tour passing through exactly one node from each cluster. An exact exponential time algorithm and an effective meta‐heuristic algorithm for the problem are presented. The meta‐heuristic proposed is a modified Ant Colony System (ACS) algorithm called reinforcing Ant Colony System which introduces new correction rules in the ACS algorithm. Computational results are reported for many standard test problems. The proposed algorithm is competitive with the other already proposed heuristics for the GTSP in both solution quality and computational time.